Cornyn, speaking to reporters on Monday, said that he was "glad" the policy — which sparked a wave of protests and constitutional challenges nationwide — was clarified by newly confirmed Homeland Security Secretary John Kelly to ensure that legal permanent residents weren't included in the temporary ban.

"They got into a big hurry," Cornyn said of the roll-out, quickly adding: "But public safety is the No. 1 responsibility of the federal government, so I certainly support the vetting of refugees coming into the country."

Cruz, who has shown continued support for Trump since endorsing him last year, blamed the public outcry on "hysteria and mistruths being pushed by the liberal media" and commended Trump "for acting swiftly to try to prevent terrorists from infiltrating our refugee programs."

The ban is a pause in refugee admissions "so that stronger vetting procedures can be put in place," he said in a statement Monday. "This is a commonsense step that the American people overwhelmingly support."

Cornyn, the Senate's No. 2 Republican leader, was among several congressional members who said they were not consulted by the White House before the order's implementation.

Trump's executive order, issued late last week, prompted spontaneous protests, strikes and a flurry of legal actions over the weekend as scores of unsuspecting travelers — including legal permanent residents — were detained and questioned in airports nationwide.

Under Trump's order, people from Iran, Syria, Yemen, Sudan, Somalia and Libya -- countries that are majority Muslim — are blocked from entering the country for 90 days. Refugees are banned for 120 days and from Syria indefinitely.

White House surrogates and some Republican lawmakers have pushed back against accusations that the ban violates the First Amendment and targets Muslims, saying it focuses on countries first identified for vetting by Obama's administration.

Cruz previously disavowed Trump's campaign call for a ban on Muslims entering the country, but has himself called for surveillance of Muslim communities in the United States, as well as a three-year moratorium on refugees from countries with a strong al-Qaeda or Islamic State influence.

Congressional leaders on both sides of the aisle were caught off guard by the order last week, forcing many to scramble to chart their own positions on the policy.

Democrats, many of whom joined protesters at airports nationwide, have since denounced the executive order as un-American and unconstitutional, while Republicans have taken more cautious, varied and nuanced approaches.

Over the weekend, some Texas Republicans quickly issued statements in support of the measure, while San Antonio Rep. Will Hurd, a former CIA officer, was the first Republican in the Texas delegation to be unequivocally opposed.

The ban is the “ultimate display of mistrust and will erode our allies' willingness to fight with us,” Hurd said in a statement Sunday, adding that it and “provides terrorists with another tool to gain sympathy and recruit new fighters."

House Homeland Security Committee Chairman Michael McCaul, R-Austin, offered praise and careful criticism of the measure over the weekend.

While noting the ensuing “confusion and uncertainty” over the ban made clear the need for adjustments, McCaul said the U.S. shouldn’t turn away people with lawful U.S. visas and green cards, particularly "those who have risked their lives serving alongside our forces overseas or who call America their home.”

House Armed Services Committee Chairman Mac Thornberry, R-Clarendon, said in a statement Monday that while it’s “understandable” the Trump would want to temporarily halt admitting refugees in order to assess the vetting process, "less understandable, however, is issuing an order with so much confusion and uncertainty that even legal permanent residents and those who have helped our troops are caught up in the chaos.”

Trump, who took to Twitter on Monday to defend his decision, has already shown willingness to target his critics — Democrats and Republicans alike — for questioning his measure. Over the weekend, the president blasted Arizona Sen. John McCain and South Carolina Sen. Lindsey Graham for criticizing the executive order.

The joint statement of former presidential candidates John McCain & Lindsey Graham is wrong - they are sadly weak on immigration. The two...

Cornyn rebuffed questions about the squabble between the Republicans, saying: “We have a lot of things we need to do, and I’m focused on the long-term and not these little battles that spring up along the way.”